\documentclass[t,12pt,aspectratio=169]{beamer} % 16:9 宽屏比例，适合现代投影
\usepackage{ctex} % 中文支持
\usepackage{amsmath, amssymb} % 数学公式与符号
\usepackage{graphicx}

% 主题设置（推荐简洁风格）
\usetheme{Madrid}
\usecolortheme{default} % 可选：seahorse, beaver, dolphin 等

\title{实变函数第二章：点集}
\author{CQX ET AL}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}

\begin{frame}
  \titlepage
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{第二章目录 }

\begin{enumerate}

\item[2.1.] 度量空间、$n$维欧氏空间
\item[2.2.] 聚点、内点、界点
\item[2.3.] 开集、闭集、完备集：开集与闭集的性质。
\item[2.4.] 直线上的开集、闭集及完备集的构造：紧集等价于有界闭集。
\item[2.5.] 康托尔三分集

\end{enumerate}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}{第二章重点 }
%
%\begin{enumerate}
%\item  开集与闭集的性质，直线上的开集、闭集与完备集的构造，$\mathbb{R}^n$ 中的开集的构造，紧集等价于有界闭集，康托尔三分集的性质。
%\item  
%\item  
%\item  
%\item  
%
%\end{enumerate}
%
%\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%\begin{frame}{第二章习题 }
%
%\begin{enumerate}
%\item  
%\item  
%\item  
%\item  
%\item  
%\item  
%
%
%\end{enumerate}
%
%\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.1.1.  度量空间、$n$维欧氏空间 }

\begin{itemize}

\item  {\color{red}问题： }
\begin{enumerate}
\item  {\color{red}什么是度量空间？}
\item  {\color{red}如何将欧氏空间看作度量空间？}

\end{enumerate}

%\item  解答：

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.1.2.  }

\begin{itemize}

\item  {\color{red}问题：}
\begin{enumerate}
\item  {\color{red}什么是欧氏空间 $\mathbb{R}^n$ 中的邻域？}
\item  {\color{red}什么是欧氏空间 $\mathbb{R}^n$ 中的收敛点列？}
\item  {\color{red}什么是欧氏空间 $\mathbb{R}^n$ 中的有界点集？}

\end{enumerate}

%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.2.1. 内点、外点、边界点、聚点、孤立点 }

\begin{itemize}

\item  {\color{red}问题：设 $E$ 是欧氏空间 $\mathbb{R}^n$ 的一个点集。什么是 $E$ 的内点、外点、边界点、聚点、孤立点？}

%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.2.2.  }

\begin{itemize}

\item  {\color{red}问题：设 $E\subseteq \mathbb{R}^n$, 设 $P_0\in E$. 证明下面的三个陈述是等价的：}
\begin{enumerate}
\item  {\color{red} $P_0$ 是 $E$ 的聚点。 }
\item  {\color{red} 在 $P_0$ 的任意邻域内，存在属于 $E$ 但不同于 $P_0$ 的点。 }
\item  {\color{red} 存在 $E$ 中互异的点 $P_n$, 使得 $P_n\to P_0$. }
\end{enumerate}

%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.2.3.  }

\begin{itemize}

\item  {\color{red}问题：设 $E\subseteq \mathbb{R}^n$, 什么是 $E$ 的开核 $\mathring{E}$、导集 $E'$、边界 $\partial E$、闭包 $\overline{E}$ ？}


%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.2.4.  }

\begin{itemize}

\item  {\color{red}问题：设 $A\subseteq B$. 证明 $A'\subseteq B'$, $\mathring{A}\subseteq \mathring{B}$, $\overline{A}\subseteq\overline{B}$. }

%\item  解答：

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.2.5.  }

\begin{itemize}

\item  {\color{red}问题：证明 $(A\cup B)' = A'\cup B'$. }


%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.2.6.  }

\begin{itemize}

\item  {\color{red}问题：设 $E\subseteq \mathbb{R}^n$ 是无限点集，但是有界的。则 $E$ 必有聚点。}

%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.2.7.  }

\begin{itemize}

\item  {\color{red}问题：设 $E\subseteq \mathbb{R}^n$ 是不是空集也不是全集。则 $E$ 必有边界点。}


%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.3.1. 开集、闭集、完备集 }

\begin{itemize}

\item  {\color{red}问题：什么是欧氏空间 $\mathbb{R}^n$ 中的开集、闭集？}


%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.3.2.  }

\begin{itemize}

\item  {\color{red}问题：设 $E\subseteq \mathbb{R}^n$, 证明 $\mathring{E}$ 是开集，$E'$ 和 $\overline{E}$ 是闭集。}


%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.3.3.  }

\begin{itemize}

\item  {\color{red}问题：}
\begin{enumerate}
\item  {\color{red}设 $E$ 是开集，则 $\mathbb{R}^n-E$ 是闭集。}
\item  {\color{red}设 $E$ 是闭集，则 $\mathbb{R}^n-E$ 是开集。}
\end{enumerate}

%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.3.4.  }

\begin{itemize}

\item  {\color{red}问题：证明：任意多个开集的并集仍是开集，有限多个开集的交集仍是开集。}


%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.3.5.  }

\begin{itemize}

\item  {\color{red}问题：证明：任意多个闭集的交集仍是闭集，有限多个闭集的并集仍是闭集。}


%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.3.6.  }

\begin{itemize}

\item  {\color{red}问题：举例说明任意多个闭集的并集不一定是闭集。}


%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.3.7.  }

\begin{itemize}

\item  {\color{red}问题：设 $F_1,F_2$ 是 $\mathbb{R}$ 中的两个互不相交的闭集。证明存在两个互不相交的开集 $G_1,G_2$ 可以分别覆盖这两个闭集，即 $F_1\subseteq G_1, F_2\subseteq G_2$. }


%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.3.8.  }

\begin{itemize}

\item  {\color{red}问题：设一族开集覆盖了一个有界闭集。证明这族开集闭定存在有限多个开集，也能覆盖这个闭集。}


%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.3.9.  }

\begin{itemize}

\item  {\color{red}问题：什么是度量空间中的紧集？}


%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.3.10.  }

\begin{itemize}

\item  {\color{red}问题：证明 $\mathbb{R}^n$ 中的紧集必定是有界闭集。}


%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.3.11.  }

\begin{itemize}

\item  {\color{red}问题：什么是 $\mathbb{R}^n$ 中的完备集？ }

%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.4.1. 直线上的开集、闭集、完备集的构造 }

\begin{itemize}

\item  {\color{red}问题：证明直线上的任意非空开集都可以写成有限个或可数个互不相交的开区间的并集。}


%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.4.2.  }

\begin{itemize}

\item  {\color{red}问题：证明直线上的闭集或者是整个直线，或者是从直线挖掉有限个或可数个互不相交的开区间所得到的余集。}


%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.5.1. 康托尔三分集 }

\begin{itemize}

\item  {\color{red}问题：什么是 $\mathbb{R}^n$ 中的稠密集、疏朗集？}


%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.5.2.  }

\begin{itemize}

\item  {\color{red}问题：什么是康托尔三分集？ }


%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.5.3.  }

\begin{itemize}

\item  {\color{red}问题：证明康托尔三分集是一个完备集、疏朗集、测度为零、但是基数仍为 $\aleph$.  }

%\item  解答：

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.5.4.  }

\begin{itemize}

\item  {\color{red}问题：什么是科赫曲线？ }


%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.5.5.  }

\begin{itemize}

\item  {\color{red}问题：什么是 Sierpinski 地毯？ }


%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.5.6.  }

\begin{itemize}

\item  {\color{red}问题：什么是分形？什么是分形的 Hausdorff 维数？ }


%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.6.1. 习题1 }

\begin{itemize}

\item  {\color{red}问题：设 $E=[0,1]\cap \mathbb{Q}$, 求 $E$ 在 $\mathbb{R}$ 中的导集、开核、闭包。 }


%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.6.2. 习题3 }

\begin{itemize}

\item  {\color{red}问题：设 
$$E=\{(x,y)\in\mathbb{R}^2 : y=\sin(1/x),x\in\mathbb{R},x\neq 0\} \cup \{(0,0) \},$$
求 $E$ 在 $\mathbb{R}^2$ 中的 $E',\mathring{E}, \overline{E}$. 

 }


%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.6.3. 习题5 }

\begin{itemize}

\item  {\color{red}问题：证明：点集 $F$ 为闭集的充分必要条件是 $F=\overline{F}$. }


%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.6.4. 习题11 }

\begin{itemize}

\item  {\color{red}问题：证明：用10进位小数表示 $[0,1]$ 中的数时，其用不着数字7的一切数组成一个完备集。 }


%\item  解答：


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.6.5. 习题12 }

\begin{itemize}

\item  {\color{red}问题：证明：函数 $f(x)$ 为 $[a,b]$ 上的连续函数的充分必要条件是对任意实数 $c$, 集合
$E_1=\{x: f(x)\ge c\}$ 与 $E_2=\{x: f(x)\le c\}$ 都是闭集。 }


%\item  解答：


\end{itemize}

\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}



